The Curvature Star Body And Its Dual Brunn-Minkowski Inequalities In The Plane

The classical Brunn-Minkowski theory originated in the doctoral thesis of H. Brunn and the pioneering work of H. Minkowski in 1887. One of the core issues of the classic Brunn-Minkowski theory is mixed volume. So the Brunn-Minkowski theory is also called as the mixed volume theory. By impelling of some world-famous mathematicians such as Bonnesen, Santalo, Fenchel, Petty, Blaschke, Hadwiger and Busemann, the Brunn-Minkowski theory has been become enough to perfect. Since the famous mathematician Lutwak introduced the concept of dual mixed volume of star body in 1975, he created a dual Brunn-Minkowski theory. It is surprisingly similar as the classic theory of convex bodies which are created by Blaschke, Alek-sandrov, Minkowski, etc. It makes the object of study converted from convex body to star body. In the 1980’s, the theory was unprecedented prosperity and solved a series of important unresolved issues of long-term.The paper is starting with an ovoid body with at least C2 smooth boundary in the plane, the reciprocal of curvature of the boundary points for the ovoid body is considered as a radial function. We construct a dual new star body, it is called as the curvature star body. The paper discusses some properties of the curvature star body. And using approach of the integral geometry and convex geometric analysis, the paper obtains some associated dual Brunn-Minkowski inequalities of the curvature star body.